Trigonometric Equations - Triple Angle Formula

The trigonometric triple-angle identities give a relationship between the basic trigonometric functions applied to three times an angle in terms of trigonometric functions of the angle itself.

Triple-angle Identities

\[\begin{align} \sin 3 \theta &= 3 \sin \theta - 4 \sin ^3 \theta \\\\ \cos 3\theta &= 4 \cos ^ 3 \theta - 3 \cos \theta \end{align}\]

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Proof: To prove the triple-angle identities, we can write \(\sin 3 \theta\) as \(\sin(2 \theta + \theta)\). Then we can use the sum formula and the double-angle identities to get the desired form:

\[\begin{align} \sin 3 \theta & = \sin (2 \theta + \theta)\\ & = \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta \\ & = (2 \sin \theta \cos \theta) \cos \theta + \big(1 - 2 \sin^2 \theta \big) \sin \theta \\ & = 2 \sin \theta \cos^2 \theta + \sin \theta - 2 \sin^3 \theta \\ & = 2 \sin \theta \big(1 - \sin^2 \theta\big) + \sin \theta - 2 \sin^3 \theta \\ & = 2 \sin \theta - 2 \sin^3 \theta + \sin \theta - 2 \sin^3 \theta \\ & = 3 \sin \theta - 4 \sin^3 \theta.\ _\square \end{align} \]

The triple-angle identity of \(\cos 3 \theta\) can be proved in a very similar manner.

From these formulas, we also have the following identities for \(\sin^3 \theta\) and \(\cos^3 \theta \) in terms of lower powers:

\[\begin{align} \sin^3 \theta &= \frac{3 \sin \theta - \sin 3\theta }{4}\\\\ \cos^3 \theta & = \frac{\cos 3\theta + 3 \cos \theta}{4}. \end{align} \]